PFE to Real, Second-Order Sections

When all coefficients of $A(z)$ and $B(z)$ are real (implying that $H(z)=B(z)/A(z)$ is a real filter), it will always happen that the complex one-pole filters will occur in complex conjugate pairs. Let $(r,p)$ denote any one-pole section in the PFE of Eq. (6.7). Then if $p$ is complex and $H(z)$ describes a real filter, we will also find $(\overline{r},\pc)$ somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:

$$\begin{eqnarray*} H(z) &=& \frac{r}{1-pz^{-1}} + \frac{\overline{r}}{1-\pc z^{-1... ...box{re}\left\{p\right\}z^{-1}+ \left\vert p\right\vert^2 z^{-2}} \end{eqnarray*}$$

Expressing the pole $p$ in polar form as $p=Re^{j\theta}$, and the residue as $r=Ge^{j\phi}$, the last expression above can be rewritten as (see §10.1.3 for details)

$$H(z) = 2G\frac{\cos(\phi)-\cos(\phi-\theta)z^{-1}}{1-2R\,\cos(\theta)z^{-1}+ R^2 z^{-2}}.$$

In other terms, every real filter with $M<N$ can be implemented as a parallel bank of biquads.^7.4However, the full generality of a biquad section is not needed, since the PFE requires only one zero per second-order section.

To see why we must stipulate $M<N$ in Eq. (6.7), consider the sum of two first-order terms by direct calculation:

$$H_2(z) = \frac{r_1}{1-p_1z^{-1}} + \frac{r_2}{1-p_2z^{-1}} = \frac{(r_1 + r_2) - (r_1 p_2 + r_2 p_1) z^{-1}}{(1-p_1z^{-1})(1-p_2z^{-1})}$$ (7.9)

Notice that the numerator order, viewed as a polynomial in $z^{-1}$, is one less than the denominator order. In the same way, it is easily shown by mathematical induction that the sum of $N$ one-pole terms $r_i/(1-p_i z^{-1})$ can produce a numerator order of at most $M=N-1$ (while the denominator order is $N$ if there are no pole-zero cancellations). Following terminology used for analog filters, we call the case $M<N$ a strictly proper transfer function.^7.5